Cluster-cyclic condition of skew-symmetrizable matrices of rank 3 via the Markov constant

In this paper, we consider mutations of skew-symmetrizable matrices of rank 3. Every skew-symmetrizable matrix corresponds to a weighted quiver, and we study the conditions when this quiver is always cyclic after applying mutations. In this study, the Markov constant has an essential meaning. It has already appeared in some previous works for skew-symmetric matrices.